and so this integral diverges. Notice that starting the integral limit at makes no difference here. The simple fact is that we know

is infinite, while the integral

is finite for any satisfying . It is really the integral on the far left, the one with the infinite limit, that causes the problem here.

Converge or Diverge?

Solution: This is really the same as the previous example. In this case, we have

and so the comparison test says that

where we already knew that the integral on the far right diverges, and the previous example showed that the integral in the middle diverges. Both with work to show that the integral on the far left diverges.

Converge or Diverge?

Solution: Notice that

simply because the smallest and largest values that can take on are and , respectively. So, we have

Of course, the side that we're interested in here is the left, since we know that the integral of from to is divergent. Thus, the integral in question diverges.

Converge or Diverge?

Solution: This one is calculated much along the same lines as the first, but the result is very different. First, we wish to have an accurate an approximation to bound the integrand by as possible. While it is true that both and have a maximum and minimum value of and respectively, we should try to find the maximum and minimum value of their sum… and since each of the individual functions has a max/min at different parts of the unit circle, the max and min of their sum will not be and , repsectively.

The first derivative of is , which has zeros at and . The second derivative is , which (by the second derivative test) tells us that has a minimum at and a maximum at , up to periodicity of the function. This allows us to state that

Which of these bounds are we actually interested in? The one of the left tells us nothing, since it only says that the integral we are considering is greater than one that is known to converge. In order to get information about convergence, the comparison test says that we much bound our integral above by one that we already know to converge… which is what the right side of the above equation does. So, the integral in question converges. (And we have relatively little idea what it may be equal to.)

Converge or Diverge?

Solution: Simplifying gives

We know already that the integral on the right converges and we can calculate it using the previous section on improper integrals. I.e., in this case we are lucky and don't need to use the comparison test at all. But, what if we had